Topos Theory

Indecomposable objects

Definition

An object XX of a category CC is indecomposable if it cannot be expressed as a non-trivial coproduct of objects of CC. Formally, XX is indecomposable if given an isomorphism X≅∐iUiX \cong \coprod_i U_i, there is a unique index ii such that X≅UiX \cong U_i and Uj≅0U_j \cong 0 for all j≠ij \neq i, where 00 is an initial object.

The requirement that ii be unique keeps the initial object itself from being indecomposable; this is analogous to being too simple to be simple.

Lambek–Scott indecomposability

Lambek & Scott give a different definition of indecomposability. Generalizing their definition slightly, we may say that an object XX is indecomposable (in the sense of Lambek–Scott) if any jointly epimorphic family{Ui→X}i\{U_i \to X\}_i of arrows into XX contains at least one epimorphismUi↠XU_i \twoheadrightarrow X, and moreover the unique arrow 0→X0 \to X is not epic (this to ensure that 00 is not indecomposable).

If the epi Ui↠XU_i \twoheadrightarrow X is required to be regular, then in an extensive category the Lambek–Scott definition implies that given above: if k:X≅∐iUik \colon X \cong \coprod_i U_i, then the family {k−1ιi:Ui→∐iUi≅X}i\{k^{-1} \iota_i \colon U_i \to \coprod_i U_i \cong X\}_i is jointly epic, so it contains a regular epi ιik−1\iota_i k^{-1}. But extensivity implies that ιi\iota_i is a monomorphism, so the regular epi ιik−1\iota_i k^{-1} is also monic and hence an isomorphism. The converse does not hold in general, but it does hold if XX is projective. See this MathOverflow thread for a discussion.

Indecomposability vs irreducibility

An indecomposable representation is precisely an indecomposable object in an appropriate category RepRep of representations, as one would expect. In contrast, an irreducible representation is precisely a simple object in RepRep. Every irreducible representation is indecomposable, but the converse holds only in special situations (such as the category of finite-dimensional linear representations of a real semisimple Lie group).

However, one level decategorified, an irreducible element? of a posetPP is precisely an indecomposable object of PP when thought of as a thin category. In contrast, a simple object is analogous to an atomic element, although they are not the same thing. (One might say that atomic = 00-simple.) Again, every atomic element is irreducible, but the converse holds only in special situations (such as the power set of any set).

The bottom line is that ‘irreducible’ and ‘indecomposable’ sometimes mean the same thing but sometimes don't, and ‘irreducible’ doesn't even mean the same thing across different fields.